86 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

1.7.5.1. Theorem (Yu). Let K be a field with positive characteristic, and A a CM

abelian variety over K with CM structure provided by a CM field L ⊂

End0(A).

If

OL ⊂ End(A) then there is a finite extension K /K such that AK equipped with

its OL-action descends to a finite field contained in K .

This result is [133, Thm. 1.3]; it will not be used in what follows. (Note that

it suﬃces just to descend the abelian variety AK to a finite subfield of K , as then

a further finite extension on K will enable us to descend the abelian variety along

with its OL-action, by Lemma 1.2.1.2.)

1.8. CM lifting questions

1.8.1. Basic definitions and examples. Let κ be a field of characteristic p 0,

and consider an abelian variety A0 over κ. By Corollary 1.6.2.5, if κ is finite and

A0 is isotypic then we may endow it with a structure of CM abelian variety having

complex multiplication by a CM field. Inspired in part by Theorem 1.6.5.1, we

wish to pose several questions related to the problem of lifting A0 to characteristic

0 in the presence of CM structures. First we make a general definition unrelated

to complex multiplication.

1.8.1.1. Definition. A lifting of A0 to characteristic 0 is a triple (R, A, φ) con-

sisting of a domain R of characteristic 0, an abelian scheme A over R, a surjective

map R κ, and an isomorphism φ : Aκ A0 of abelian varieties over κ.

We may replace R with its localization at the maximal ideal ker(R κ) so

that R is local with residue field κ. For K := Frac(R), if AK admits suﬃciently

many complex multiplications then we say A is a CM lift of A0 to characteristic 0.

The injective map End(A) → End(AK ) has torsion-free cokernel:

1.8.2. Lemma. For abelian schemes A, B over an integral scheme S with generic

point η, the injective map Hom(A, B) → Hom(Aη,Bη) has torsion-free cokernel.

Proof. Consider f : Aη → Bη such that n · f extends to an S-group map

h : A → B for a non-zero integer n. The restriction h : A[n] → B[n] between finite

flat S-groups vanishes because such vanishing holds on the generic fiber over the

integral S. Since [n] : A → A is an fppf covering with kernel A[n], it follows that h

factors through this map over S, which is to say h = n · f for some S-group map

f : A → B. Hence, the map fη − f ∈ Hom(Aη,Bη) is killed by n, so fη = f.

The injective map in Lemma 1.8.2 can fail to be surjective:

1.8.3. Example. Let p be a prime with p ≡ 3 (mod 4), so p is prime in Z[i] (with

i2

= −1). Let R = Z(p) +pZ(p)[i], so [Z(p)[i] : R] = p and Frac(R) = Q(i). Let E be

the elliptic curve y2 = x3 − x over R, so the generic fiber EQ(i) has endomorphism

ring Z[i] via the action [i](x, y) = (−x, −iy). Because [i]∗(dx/y) = i · dx/y, Z[i]

acts on Lie(EQ(i)) through scaling via the canonical inclusion Z[i] → Q(i).

We claim that End(E) = Z (so

End0(E)

:= Q⊗Z End(E) = Q, even though the

generic fiber EQ(i) has endomorphism algebra Q(i)). Indeed, if not then End(E) is

an order in Z[i] = End(EQ(i)), so End(E) = Z[i] by Lemma 1.8.2. In particular,

the action by i on EQ(i) would extend to an action on E, so the resulting multiplier